Showing papers on "Partial differential equation published in 2014"

A theory of regularity structures

Abstract: We introduce a new notion of “regularity structure” that provides an algebraic framework allowing to describe functions and/or distributions via a kind of “jet” or local Taylor expansion around each point. The main novel idea is to replace the classical polynomial model which is suitable for describing smooth functions by arbitrary models that are purpose-built for the problem at hand. In particular, this allows to describe the local behaviour not only of functions but also of large classes of distributions. We then build a calculus allowing to perform the various operations (multiplication, composition with smooth functions, integration against singular kernels) necessary to formulate fixed point equations for a very large class of semilinear PDEs driven by some very singular (typically random) input. This allows, for the first time, to give a mathematically rigorous meaning to many interesting stochastic PDEs arising in physics. The theory comes with convergence results that allow to interpret the solutions obtained in this way as limits of classical solutions to regularised problems, possibly modified by the addition of diverging counterterms. These counterterms arise naturally through the action of a “renormalisation group” which is defined canonically in terms of the regularity structure associated to the given class of PDEs. Our theory also allows to easily recover many existing results on singular stochastic PDEs (KPZ equation, stochastic quantisation equations, Burgers-type equations) and to understand them as particular instances of a unified framework. One surprising insight is that in all of these instances local solutions are actually “smooth” in the sense that they can be approximated locally to arbitrarily high degree as linear combinations of a fixed family of random functions/distributions that play the role of “polynomials” in the theory. As an example of a novel application, we solve the long-standing problem of building a natural Markov process that is symmetric with respect to the (finite volume) measure describing the $$\Phi ^4_3$$ Euclidean quantum field theory. It is natural to conjecture that the Markov process built in this way describes the Glauber dynamic of $$3$$ -dimensional ferromagnets near their critical temperature.

Numerical methods for kinetic equations

Abstract: In this survey we consider the development and mathematical analysis of numerical methods for kinetic partial differential equations. Kinetic equations represent a way of describing the time evolution of a system consisting of a large number of particles. Due to the high number of dimensions and their intrinsic physical properties, the construction of numerical methods represents a challenge and requires a careful balance between accuracy and computational complexity. Here we review the basic numerical techniques for dealing with such equations, including the case of semi-Lagrangian methods, discrete-velocity models and spectral methods. In addition we give an overview of the current state of the art of numerical methods for kinetic equations. This covers the derivation of fast algorithms, the notion of asymptotic-preserving methods and the construction of hybrid schemes.

Unified form language: A domain-specific language for weak formulations of partial differential equations

Abstract: We present the Unified Form Language (UFL), which is a domain-specific language for representing weak formulations of partial differential equations with a view to numerical approximation. Features of UFL include support for variational forms and functionals, automatic differentiation of forms and expressions, arbitrary function space hierarchies for multifield problems, general differential operators and flexible tensor algebra. With these features, UFL has been used to effortlessly express finite element methods for complex systems of partial differential equations in near-mathematical notation, resulting in compact, intuitive and readable programs. We present in this work the language and its construction. An implementation of UFL is freely available as an open-source software library. The library generates abstract syntax tree representations of variational problems, which are used by other software libraries to generate concrete low-level implementations. Some application examples are presented and libraries that support UFL are highlighted.

Mean Field Games Models—A Brief Survey

Abstract: The mean-field framework was developed to study systems with an infinite number of rational agents in competition, which arise naturally in many applications. The systematic study of these problems was started, in the mathematical community by Lasry and Lions, and independently around the same time in the engineering community by P. Caines, Minyi Huang, and Roland Malhame. Since these seminal contributions, the research in mean-field games has grown exponentially, and in this paper we present a brief survey of mean-field models as well as recent results and techniques. In the first part of this paper, we study reduced mean-field games, that is, mean-field games, which are written as a system of a Hamilton–Jacobi equation and a transport or Fokker–Planck equation. We start by the derivation of the models and by describing some of the existence results available in the literature. Then we discuss the uniqueness of a solution and propose a definition of relaxed solution for mean-field games that allows to establish uniqueness under minimal regularity hypothesis. A special class of mean-field games that we discuss in some detail is equivalent to the Euler–Lagrange equation of suitable functionals. We present in detail various additional examples, including extensions to population dynamics models. This section ends with a brief overview of the random variables point of view as well as some applications to extended mean-field games models. These extended models arise in problems where the costs incurred by the agents depend not only on the distribution of the other agents, but also on their actions. The second part of the paper concerns mean-field games in master form. These mean-field games can be modeled as a partial differential equation in an infinite dimensional space. We discuss both deterministic models as well as problems where the agents are correlated. We end the paper with a mean-field model for price impact.

An Introduction to Stochastic Differential Equations

Abstract: This book provides a quick, but very readable introduction to stochastic differential equations-that is, to differential equations subject to additive "white noise" and related random disturbances. The exposition is strongly focused upon the interplay between probabilistic intuition and mathematical rigour. Topics include a quick survey of measure theoretic probability theory, followed by an introduction to Brownian motion and the Ito stochastic calculus, and finally the theory of stochastic differential equations. The text also includes applications to partial differential equations, optimal stopping problems and options pricing.

Adaptive Partial Differential Equation Observer for Battery State-of-Charge/State-of-Health Estimation Via an Electrochemical Model

Abstract: This paper develops an adaptive partial differential equation (PDE) observer for battery state-of-charge (SOC) and state-of-health (SOH) estimation. Real-time state and parameter information enables operation near physical limits without compromising durability, thereby unlocking the full potential of battery energy storage. SOC/SOH estimation is technically challenging because battery dynamics are governed by electrochemical principles, mathematically modeled by PDEs. We cast this problem as a simultaneous state (SOC) and parameter (SOH) estimation design for a linear PDE with a nonlinear output mapping. Several new theoretical ideas are developed, integrated together, and tested. These include a backstepping PDE state estimator, a Pade-based parameter identifier, nonlinear parameter sensitivity analysis, and adaptive inversion of nonlinear output functions. The key novelty of this design is a combined SOC/SOH battery estimation algorithm that identifies physical system variables, from measurements of voltage and current only.

Application of first integral method to fractional partial differential equations

Abstract: In this paper, fractional derivatives in the sense of modified Riemann-Liouville derivative and first integral method are applied for constructing exact solutions of nonlinear fractional generalized reaction duffing model and nonlinear fractional diffusion reaction equation with quadratic and cubic nonlinearity. Our approach provides first integrals in polynomial form with a high accuracy for two-dimensional plane autonomous systems. Exact soliton solutions are constructed through established first integrals.

Partial differential equation models in macroeconomics.

Abstract: The purpose of this article is to get mathematicians interested in studying a number of partial differential equations (PDEs) that naturally arise in macroeconomics These PDEs come from models designed to study some of the most important questions in economics At the same time, they are highly interesting for mathematicians because their structure is often quite difficult We present a number of examples of such PDEs, discuss what is known about their properties, and list some open questions for future research

In-plane and out-of-plane motion characteristics of microbeams with modal interactions

Abstract: The three-dimensional nonlinear size-dependent motion characteristics of a microbeam are investigated numerically, with special consideration to one-to-one internal resonances between the in-plane and out-of-plane transverse modes. All of the in-plane and out-of-plane displacements and inertia are taken into account and Hamilton’s principle, in conjunction with the modified couple stress theory, is employed to obtain the nonlinear partial differential equations governing the motions of the system in the in-plane and out-of-plane directions. The discretization procedure is carried out by applying the Galerkin technique to the partial differential equations of motion, yielding a set of nonlinear ordinary differential equations. A linear analysis is performed upon this set of equations so as to obtain the size-dependent natural frequencies of the system. The nonlinear analysis of the discretized equations of motion is carried out by employing the pseudo-arclength continuation technique, resulting in the resonant responses of the system. It is shown that, due to the presence of one-to-one internal resonances between the in-plane and out-of-plane transverse modes, an in-plane excitation can give rise to an out-of-plane displacement; the internal resonances also cause the occurrence of extra solution branches and new bifurcation points.

Fuzzy Boundary Control Design for a Class of Nonlinear Parabolic Distributed Parameter Systems

Abstract: This paper deals with the problem of fuzzy boundary control design for a class of nonlinear distributed parameter systems which are described by semilinear parabolic partial differential equations (PDEs). Both distributed measurement form and collocated boundary measurement form are considered. A Takagi–Sugeno (T–S) fuzzy PDE model is first applied to accurately represent the semilinear parabolic PDE system. Based on the T–S fuzzy PDE model, two types of fuzzy boundary controllers, which are easily implemented since only boundary actuators are used, are proposed to ensure the exponential stability of the resulting closed-loop system. Sufficient conditions of exponential stabilization are established by employing the Lyapunov direct method and the vector-valued Wirtinger's inequality and presented in terms of standard linear matrix inequalities. Finally, the advantages and effectiveness of the proposed control methodology are demonstrated by the simulation results of two examples.

Continuity equations and ODE flows with non-smooth velocity*

Abstract: In this paper we review many aspects of the well-posedness theory for the Cauchy problem for the continuity and transport equations and for the ordinary differential equation (ODE). In this framework, we deal with velocity fields that are not smooth, but enjoy suitable ‘weak differentiability’ assumptions. We first explore the connection between the partial differential equation (PDE) and the ODE in a very general non-smooth setting. Then we address the renormalization property for the PDE and prove that such a property holds for Sobolev velocity fields and for bounded variation velocity fields. Finally, we present an approach to the ODE theory based on quantitative estimates.

A Cahn–Hilliard-type phase-field theory for species diffusion coupled with large elastic deformations: Application to phase-separating Li-ion electrode materials

Abstract: We formulate a unified framework of balance laws and thermodynamically-consistent constitutive equations which couple Cahn–Hilliard-type species diffusion with large elastic deformations of a body. The traditional Cahn–Hilliard theory, which is based on the species concentration c and its spatial gradient ∇ c , leads to a partial differential equation for the concentration which involves fourth-order spatial derivatives in c; this necessitates use of basis functions in finite-element solution procedures that are piecewise smooth and globally C 1 - continuous . In order to use standard C 0 - continuous finite-elements to implement our phase-field model, we use a split-method to reduce the fourth-order equation into two second-order partial differential equations (pdes). These two pdes, when taken together with the pde representing the balance of forces, represent the three governing pdes for chemo-mechanically coupled problems. These are amenable to finite-element solution methods which employ standard C 0 - continuous finite-element basis functions. We have numerically implemented our theory by writing a user-element subroutine for the widely used finite-element program Abaqus/Standard. We use this numerically implemented theory to first study the diffusion-only problem of spinodal decomposition in the absence of any mechanical deformation. Next, we use our fully coupled theory and numerical-implementation to study the combined effects of diffusion and stress on the lithiation of a representative spheroidal-shaped particle of a phase-separating electrode material.

Numerical Methods for the Fractional Laplacian: a Finite Difference-quadrature Approach

Abstract: The fractional Laplacian $(-\Delta)^{\alpha/2}$ is a nonlocal operator which depends on the parameter $\alpha$ and recovers the usual Laplacian as $\alpha \to 2$. A numerical method for the fractional Laplacian is proposed, based on the singular integral representation for the operator. The method combines finite differences with numerical quadrature to obtain a discrete convolution operator with positive weights. The accuracy of the method is shown to be $O(h^{3-\alpha})$. Convergence of the method is proven. The treatment of far field boundary conditions using an asymptotic approximation to the integral is used to obtain an accurate method. Numerical experiments on known exact solutions validate the predicted convergence rates. Computational examples include exponentially and algebraically decaying solutions with varying regularity. The generalization to nonlinear equations involving the operator is discussed: the obstacle problem for the fractional Laplacian is computed.

Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equations

Abstract: We prove some new Strichartz estimates for a class of dispersive equations with radial initial data. In particular, we obtain the full radial Strichartz estimates up to some endpoints for the Schrodinger equation. Using these estimates, we obtain some new results related to nonlinear problems, including small data scattering and large data LWP for the nonlinear Schrodinger and wave equations with radial critical initial data and the well-posedness theory for the fractional order Schrodinger equation in the radial case.

Second-order ( 2 + 1 ) -dimensional anisotropic hydrodynamics

Abstract: We present a complete formulation of second-order $(2+1)$-dimensional anisotropic hydrodynamics. The resulting framework generalizes leading-order anisotropic hydrodynamics by allowing for deviations of the one-particle distribution function from the spheroidal form assumed at leading order. We derive complete second-order equations of motion for the additional terms in the macroscopic currents generated by these deviations from their kinetic definition using a Grad-Israel-Stewart 14-moment ansatz. The result is a set of coupled partial differential equations for the momentum-space anisotropy parameter, effective temperature, the transverse components of the fluid four-velocity, and the viscous tensor components generated by deviations of the distribution from spheroidal form. We then perform a quantitative test of our approach by applying it to the case of one-dimensional boost-invariant expansion in the relaxation time approximation (RTA) in which case it is possible to numerically solve the Boltzmann equation exactly. We demonstrate that the second-order anisotropic hydrodynamics approach provides an excellent approximation to the exact (0+1)-dimensional RTA solution for both small and large values of the shear viscosity.

Time-optimal path planning in dynamic flows using level set equations: theory and schemes

Abstract: We develop an accurate partial differential equation-based methodology that predicts the time-optimal paths of autonomous vehicles navigating in any continuous, strong, and dynamic ocean currents, obviating the need for heuristics. The goal is to predict a sequence of steering directions so that vehicles can best utilize or avoid currents to minimize their travel time. Inspired by the level set method, we derive and demonstrate that a modified level set equation governs the time-optimal path in any continuous flow. We show that our algorithm is computationally efficient and apply it to a number of experiments. First, we validate our approach through a simple benchmark application in a Rankine vortex flow for which an analytical solution is available. Next, we apply our methodology to more complex, simulated flow fields such as unsteady double-gyre flows driven by wind stress and flows behind a circular island. These examples show that time-optimal paths for multiple vehicles can be planned even in the presence of complex flows in domains with obstacles. Finally, we present and support through illustrations several remarks that describe specific features of our methodology.

Heat and mass transfer in unsteady rotating fluid flow with binary chemical reaction and activation energy.

Abstract: In this study, the Spectral Relaxation Method (SRM) is used to solve the coupled highly nonlinear system of partial differential equations due to an unsteady flow over a stretching surface in an incompressible rotating viscous fluid in presence of binary chemical reaction and Arrhenius activation energy. The velocity, temperature and concentration distributions as well as the skin-friction, heat and mass transfer coefficients have been obtained and discussed for various physical parametric values. The numerical results obtained by (SRM) are then presented graphically and discussed to highlight the physical implications of the simulations.

The numerical solution of nonlinear high dimensional generalized Benjamin-Bona-Mahony-Burgers equation via the meshless method of radial basis functions

Abstract: In this paper a numerical technique is proposed for solving the nonlinear generalized Benjamin-Bona-Mahony-Burgers equation. Firstly, we obtain a time discrete scheme by approximating the first-order time derivative via forward finite difference formula, then we use Kansa's approach to approximate the spatial derivatives. We prove that the time discrete scheme is unconditionally stable and convergent in time variable using the energy method. Also, we show that convergence order of the time discrete scheme is O(@t). We solve the two-dimensional version of this equation using the method presented in this paper on different geometries such as the rectangular, triangular and circular domains and also the three-dimensional case is solved on the cubical and spherical domains. The aim of this paper is to show that the meshless method based on the radial basis functions and collocation approach is also suitable for the treatment of the nonlinear partial differential equations. Also, several test problems including the three-dimensional case are given. Numerical examples confirm the efficiency of the proposed scheme.

Optimal error estimates of Galerkin finite element methods for stochastic partial differential equations with multiplicative noise

Abstract: We consider Galerkin finite element methods for semilinear sto- chastic partial differential equations (SPDEs) with multiplicative noise and Lipschitz continuous nonlinearities We analyze the strong error of conver- gence for spatially semidiscrete approximations as well as a spatio-temporal discretization which is based on a linear implicit Euler-Maruyama method In both cases we obtain optimal error estimates The proofs are based on sharp integral versions of well-known error es- timates for the corresponding deterministic linear homogeneous equation to- gether with optimal regularity results for the mild solution of the SPDE The results hold for different Galerkin methods such as the standard finite element method or spectral Galerkin approximations

The Master Equation for Large Population Equilibriums

Abstract: We use a simple \(N\)-player stochastic game with idiosyncratic and common noises to introduce the concept of Master Equation originally proposed by Lions in his lectures at the College de France. Controlling the limit \(N\rightarrow \infty \) of the explicit solution of the \(N\)-player game, we highlight the stochastic nature of the limit distributions of the states of the players due to the fact that the random environment does not average out in the limit, and we recast the Mean Field Game (MFG) paradigm in a set of coupled Stochastic Partial Differential Equations (SPDEs). The first one is a forward stochastic Kolmogorov equation giving the evolution of the conditional distributions of the states of the players given the common noise. The second is a form of stochastic Hamilton Jacobi Bellman (HJB) equation providing the solution of the optimization problem when the flow of conditional distributions is given. Being highly coupled, the system reads as an infinite dimensional Forward Backward Stochastic Differential Equation (FBSDE). Uniqueness of a solution and its Markov property lead to the representation of the solution of the backward equation (i.e. the value function of the stochastic HJB equation) as a deterministic function of the solution of the forward Kolmogorov equation, function which is usually called the decoupling field of the FBSDE. The (infinite dimensional) PDE satisfied by this decoupling field is identified with the master equation. We also show that this equation can be derived for other large populations equilibriums like those given by the optimal control of McKean-Vlasov stochastic differential equations. The paper is written more in the style of a review than a technical paper, and we spend more time motivating and explaining the probabilistic interpretation of the Master Equation, than identifying the most general set of assumptions under which our claims are true.

The Defocusing NLS Equation and Its Normal Form

Abstract: The theme of this monograph is the nonlinear Schrodinger equation. This equation models slowly varying wave envelopes in dispersive media and arises in various physical systems such as water waves, plasma physics , solid state physics and nonlinear optics. More specifically, we consider the defocusing nonlinear Schrodinger (dNLS) equation in one space dimension, iut = - uxx + 2|u|^2u, with periodic boundary conditions. With a viewpoint from infinite dimensional Hamiltonian systems we present a concise and self-contained study of this evolution equation. By developing its normal form theory we show that it is an integrable partial differential equation (PDE) in the strongest possible sense: action--angle coordinates can be constructed which lead to a globally defined coordinate system where the Hamiltonian of the dNLS equation is a function of the actions alone. Actually, this coordinate system simultaneously works for all the Hamiltonians in the dNLS hierarchy. As an immediate consequence it follows that all solutions of the dNLS equation on the circle are periodic, quasi-periodic or almost-periodic in time. Most importantly, such a coordinate system can be used to analyze qualitative properties of the solutions and to study Hamiltonian perturbations of this equation. The book is not only intended for the handful specialists working at the intersection of integrable PDEs and dynamical systems, but also researchers farther away from these fields. In fact, with the aim of reaching out to graduate students as well we have made the book self-containded. In particular we present a detailed study of the spectral theory of (near) self-adjoint Zakharov--Shabat operators on an interval which appear in the Lax pair formulation of the dNLS equation. It is key to the normal form theory of this integrable PDE. Furthermore, the book is written in a modular fashion where each of its chapters as well as its appendices may be read independently of each other.